Step |
Hyp |
Ref |
Expression |
1 |
|
weiun.1 |
⊢ 𝐹 = ( 𝑤 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ ( ℩ 𝑢 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵 } ∀ 𝑣 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵 } ¬ 𝑣 𝑅 𝑢 ) ) |
2 |
|
weiun.2 |
⊢ 𝑇 = { 〈 𝑦 , 𝑧 〉 ∣ ( ( 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ∧ ( ( 𝐹 ‘ 𝑦 ) 𝑅 ( 𝐹 ‘ 𝑧 ) ∨ ( ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑧 ) ∧ 𝑦 ⦋ ( 𝐹 ‘ 𝑦 ) / 𝑥 ⦌ 𝑆 𝑧 ) ) ) } |
3 |
|
weiunlem2.3 |
⊢ ( 𝜑 → 𝑅 We 𝐴 ) |
4 |
|
weiunlem2.4 |
⊢ ( 𝜑 → 𝑅 Se 𝐴 ) |
5 |
|
riotaex |
⊢ ( ℩ 𝑢 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵 } ∀ 𝑣 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵 } ¬ 𝑣 𝑅 𝑢 ) ∈ V |
6 |
5 1
|
fnmpti |
⊢ 𝐹 Fn ∪ 𝑥 ∈ 𝐴 𝐵 |
7 |
6
|
a1i |
⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → 𝐹 Fn ∪ 𝑥 ∈ 𝐴 𝐵 ) |
8 |
|
breq2 |
⊢ ( 𝑢 = 𝑟 → ( 𝑣 𝑅 𝑢 ↔ 𝑣 𝑅 𝑟 ) ) |
9 |
8
|
notbid |
⊢ ( 𝑢 = 𝑟 → ( ¬ 𝑣 𝑅 𝑢 ↔ ¬ 𝑣 𝑅 𝑟 ) ) |
10 |
9
|
ralbidv |
⊢ ( 𝑢 = 𝑟 → ( ∀ 𝑣 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵 } ¬ 𝑣 𝑅 𝑢 ↔ ∀ 𝑣 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵 } ¬ 𝑣 𝑅 𝑟 ) ) |
11 |
10
|
cbvriotavw |
⊢ ( ℩ 𝑢 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵 } ∀ 𝑣 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵 } ¬ 𝑣 𝑅 𝑢 ) = ( ℩ 𝑟 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵 } ∀ 𝑣 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵 } ¬ 𝑣 𝑅 𝑟 ) |
12 |
|
eleq1w |
⊢ ( 𝑤 = 𝑡 → ( 𝑤 ∈ 𝐵 ↔ 𝑡 ∈ 𝐵 ) ) |
13 |
12
|
rabbidv |
⊢ ( 𝑤 = 𝑡 → { 𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵 } = { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ) |
14 |
|
breq1 |
⊢ ( 𝑣 = 𝑠 → ( 𝑣 𝑅 𝑟 ↔ 𝑠 𝑅 𝑟 ) ) |
15 |
14
|
notbid |
⊢ ( 𝑣 = 𝑠 → ( ¬ 𝑣 𝑅 𝑟 ↔ ¬ 𝑠 𝑅 𝑟 ) ) |
16 |
15
|
cbvralvw |
⊢ ( ∀ 𝑣 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵 } ¬ 𝑣 𝑅 𝑟 ↔ ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵 } ¬ 𝑠 𝑅 𝑟 ) |
17 |
13
|
raleqdv |
⊢ ( 𝑤 = 𝑡 → ( ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵 } ¬ 𝑠 𝑅 𝑟 ↔ ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ¬ 𝑠 𝑅 𝑟 ) ) |
18 |
16 17
|
bitrid |
⊢ ( 𝑤 = 𝑡 → ( ∀ 𝑣 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵 } ¬ 𝑣 𝑅 𝑟 ↔ ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ¬ 𝑠 𝑅 𝑟 ) ) |
19 |
13 18
|
riotaeqbidv |
⊢ ( 𝑤 = 𝑡 → ( ℩ 𝑟 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵 } ∀ 𝑣 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵 } ¬ 𝑣 𝑅 𝑟 ) = ( ℩ 𝑟 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ¬ 𝑠 𝑅 𝑟 ) ) |
20 |
11 19
|
eqtrid |
⊢ ( 𝑤 = 𝑡 → ( ℩ 𝑢 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵 } ∀ 𝑣 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵 } ¬ 𝑣 𝑅 𝑢 ) = ( ℩ 𝑟 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ¬ 𝑠 𝑅 𝑟 ) ) |
21 |
20 1 5
|
fvmpt3i |
⊢ ( 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 → ( 𝐹 ‘ 𝑡 ) = ( ℩ 𝑟 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ¬ 𝑠 𝑅 𝑟 ) ) |
22 |
21
|
adantl |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) → ( 𝐹 ‘ 𝑡 ) = ( ℩ 𝑟 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ¬ 𝑠 𝑅 𝑟 ) ) |
23 |
|
eliun |
⊢ ( 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃ 𝑥 ∈ 𝐴 𝑡 ∈ 𝐵 ) |
24 |
|
rabn0 |
⊢ ( { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ≠ ∅ ↔ ∃ 𝑥 ∈ 𝐴 𝑡 ∈ 𝐵 ) |
25 |
23 24
|
bitr4i |
⊢ ( 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ≠ ∅ ) |
26 |
|
ssrab2 |
⊢ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ⊆ 𝐴 |
27 |
|
wereu2 |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ⊆ 𝐴 ∧ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ≠ ∅ ) ) → ∃! 𝑟 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ¬ 𝑠 𝑅 𝑟 ) |
28 |
26 27
|
mpanr1 |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ≠ ∅ ) → ∃! 𝑟 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ¬ 𝑠 𝑅 𝑟 ) |
29 |
25 28
|
sylan2b |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) → ∃! 𝑟 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ¬ 𝑠 𝑅 𝑟 ) |
30 |
|
riotacl2 |
⊢ ( ∃! 𝑟 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ¬ 𝑠 𝑅 𝑟 → ( ℩ 𝑟 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ¬ 𝑠 𝑅 𝑟 ) ∈ { 𝑟 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ∣ ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ¬ 𝑠 𝑅 𝑟 } ) |
31 |
29 30
|
syl |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) → ( ℩ 𝑟 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ¬ 𝑠 𝑅 𝑟 ) ∈ { 𝑟 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ∣ ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ¬ 𝑠 𝑅 𝑟 } ) |
32 |
22 31
|
eqeltrd |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) → ( 𝐹 ‘ 𝑡 ) ∈ { 𝑟 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ∣ ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ¬ 𝑠 𝑅 𝑟 } ) |
33 |
|
elrabi |
⊢ ( ( 𝐹 ‘ 𝑡 ) ∈ { 𝑟 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ∣ ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ¬ 𝑠 𝑅 𝑟 } → ( 𝐹 ‘ 𝑡 ) ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ) |
34 |
|
elrabi |
⊢ ( ( 𝐹 ‘ 𝑡 ) ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } → ( 𝐹 ‘ 𝑡 ) ∈ 𝐴 ) |
35 |
32 33 34
|
3syl |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) → ( 𝐹 ‘ 𝑡 ) ∈ 𝐴 ) |
36 |
35
|
ralrimiva |
⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → ∀ 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ( 𝐹 ‘ 𝑡 ) ∈ 𝐴 ) |
37 |
|
ffnfv |
⊢ ( 𝐹 : ∪ 𝑥 ∈ 𝐴 𝐵 ⟶ 𝐴 ↔ ( 𝐹 Fn ∪ 𝑥 ∈ 𝐴 𝐵 ∧ ∀ 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ( 𝐹 ‘ 𝑡 ) ∈ 𝐴 ) ) |
38 |
7 36 37
|
sylanbrc |
⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → 𝐹 : ∪ 𝑥 ∈ 𝐴 𝐵 ⟶ 𝐴 ) |
39 |
|
dfsbcq |
⊢ ( 𝑠 = ( 𝐹 ‘ 𝑡 ) → ( [ 𝑠 / 𝑥 ] 𝑡 ∈ 𝐵 ↔ [ ( 𝐹 ‘ 𝑡 ) / 𝑥 ] 𝑡 ∈ 𝐵 ) ) |
40 |
|
nfcv |
⊢ Ⅎ 𝑥 𝐴 |
41 |
40
|
elrabsf |
⊢ ( 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ↔ ( 𝑠 ∈ 𝐴 ∧ [ 𝑠 / 𝑥 ] 𝑡 ∈ 𝐵 ) ) |
42 |
41
|
simprbi |
⊢ ( 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } → [ 𝑠 / 𝑥 ] 𝑡 ∈ 𝐵 ) |
43 |
39 42
|
vtoclga |
⊢ ( ( 𝐹 ‘ 𝑡 ) ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } → [ ( 𝐹 ‘ 𝑡 ) / 𝑥 ] 𝑡 ∈ 𝐵 ) |
44 |
32 33 43
|
3syl |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) → [ ( 𝐹 ‘ 𝑡 ) / 𝑥 ] 𝑡 ∈ 𝐵 ) |
45 |
|
sbcel2 |
⊢ ( [ ( 𝐹 ‘ 𝑡 ) / 𝑥 ] 𝑡 ∈ 𝐵 ↔ 𝑡 ∈ ⦋ ( 𝐹 ‘ 𝑡 ) / 𝑥 ⦌ 𝐵 ) |
46 |
44 45
|
sylib |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) → 𝑡 ∈ ⦋ ( 𝐹 ‘ 𝑡 ) / 𝑥 ⦌ 𝐵 ) |
47 |
46
|
ralrimiva |
⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → ∀ 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 𝑡 ∈ ⦋ ( 𝐹 ‘ 𝑡 ) / 𝑥 ⦌ 𝐵 ) |
48 |
|
simpr |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( 𝑠 ∈ 𝐴 ∧ 𝑡 ∈ ⦋ 𝑠 / 𝑥 ⦌ 𝐵 ) ) → ( 𝑠 ∈ 𝐴 ∧ 𝑡 ∈ ⦋ 𝑠 / 𝑥 ⦌ 𝐵 ) ) |
49 |
|
sbcel2 |
⊢ ( [ 𝑠 / 𝑥 ] 𝑡 ∈ 𝐵 ↔ 𝑡 ∈ ⦋ 𝑠 / 𝑥 ⦌ 𝐵 ) |
50 |
49
|
anbi2i |
⊢ ( ( 𝑠 ∈ 𝐴 ∧ [ 𝑠 / 𝑥 ] 𝑡 ∈ 𝐵 ) ↔ ( 𝑠 ∈ 𝐴 ∧ 𝑡 ∈ ⦋ 𝑠 / 𝑥 ⦌ 𝐵 ) ) |
51 |
41 50
|
bitri |
⊢ ( 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ↔ ( 𝑠 ∈ 𝐴 ∧ 𝑡 ∈ ⦋ 𝑠 / 𝑥 ⦌ 𝐵 ) ) |
52 |
48 51
|
sylibr |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( 𝑠 ∈ 𝐴 ∧ 𝑡 ∈ ⦋ 𝑠 / 𝑥 ⦌ 𝐵 ) ) → 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ) |
53 |
52
|
ne0d |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( 𝑠 ∈ 𝐴 ∧ 𝑡 ∈ ⦋ 𝑠 / 𝑥 ⦌ 𝐵 ) ) → { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ≠ ∅ ) |
54 |
53 25
|
sylibr |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( 𝑠 ∈ 𝐴 ∧ 𝑡 ∈ ⦋ 𝑠 / 𝑥 ⦌ 𝐵 ) ) → 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) |
55 |
|
breq2 |
⊢ ( 𝑟 = ( 𝐹 ‘ 𝑡 ) → ( 𝑠 𝑅 𝑟 ↔ 𝑠 𝑅 ( 𝐹 ‘ 𝑡 ) ) ) |
56 |
55
|
notbid |
⊢ ( 𝑟 = ( 𝐹 ‘ 𝑡 ) → ( ¬ 𝑠 𝑅 𝑟 ↔ ¬ 𝑠 𝑅 ( 𝐹 ‘ 𝑡 ) ) ) |
57 |
56
|
ralbidv |
⊢ ( 𝑟 = ( 𝐹 ‘ 𝑡 ) → ( ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ¬ 𝑠 𝑅 𝑟 ↔ ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ¬ 𝑠 𝑅 ( 𝐹 ‘ 𝑡 ) ) ) |
58 |
57
|
elrab |
⊢ ( ( 𝐹 ‘ 𝑡 ) ∈ { 𝑟 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ∣ ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ¬ 𝑠 𝑅 𝑟 } ↔ ( ( 𝐹 ‘ 𝑡 ) ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ∧ ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ¬ 𝑠 𝑅 ( 𝐹 ‘ 𝑡 ) ) ) |
59 |
58
|
simprbi |
⊢ ( ( 𝐹 ‘ 𝑡 ) ∈ { 𝑟 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ∣ ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ¬ 𝑠 𝑅 𝑟 } → ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ¬ 𝑠 𝑅 ( 𝐹 ‘ 𝑡 ) ) |
60 |
32 59
|
syl |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) → ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ¬ 𝑠 𝑅 ( 𝐹 ‘ 𝑡 ) ) |
61 |
54 60
|
syldan |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( 𝑠 ∈ 𝐴 ∧ 𝑡 ∈ ⦋ 𝑠 / 𝑥 ⦌ 𝐵 ) ) → ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ¬ 𝑠 𝑅 ( 𝐹 ‘ 𝑡 ) ) |
62 |
|
rsp |
⊢ ( ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ¬ 𝑠 𝑅 ( 𝐹 ‘ 𝑡 ) → ( 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } → ¬ 𝑠 𝑅 ( 𝐹 ‘ 𝑡 ) ) ) |
63 |
61 52 62
|
sylc |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( 𝑠 ∈ 𝐴 ∧ 𝑡 ∈ ⦋ 𝑠 / 𝑥 ⦌ 𝐵 ) ) → ¬ 𝑠 𝑅 ( 𝐹 ‘ 𝑡 ) ) |
64 |
63
|
ralrimivva |
⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → ∀ 𝑠 ∈ 𝐴 ∀ 𝑡 ∈ ⦋ 𝑠 / 𝑥 ⦌ 𝐵 ¬ 𝑠 𝑅 ( 𝐹 ‘ 𝑡 ) ) |
65 |
38 47 64
|
3jca |
⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → ( 𝐹 : ∪ 𝑥 ∈ 𝐴 𝐵 ⟶ 𝐴 ∧ ∀ 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 𝑡 ∈ ⦋ ( 𝐹 ‘ 𝑡 ) / 𝑥 ⦌ 𝐵 ∧ ∀ 𝑠 ∈ 𝐴 ∀ 𝑡 ∈ ⦋ 𝑠 / 𝑥 ⦌ 𝐵 ¬ 𝑠 𝑅 ( 𝐹 ‘ 𝑡 ) ) ) |
66 |
3 4 65
|
syl2anc |
⊢ ( 𝜑 → ( 𝐹 : ∪ 𝑥 ∈ 𝐴 𝐵 ⟶ 𝐴 ∧ ∀ 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 𝑡 ∈ ⦋ ( 𝐹 ‘ 𝑡 ) / 𝑥 ⦌ 𝐵 ∧ ∀ 𝑠 ∈ 𝐴 ∀ 𝑡 ∈ ⦋ 𝑠 / 𝑥 ⦌ 𝐵 ¬ 𝑠 𝑅 ( 𝐹 ‘ 𝑡 ) ) ) |